## Distributive Property Discussions

My fifth grade class is in the middle of a unit on pre-algebra concepts.  We explored the different associative, commutative and distributive properties earlier in the week.  Students were able to use the first two with modest success, but the distributive property was causing some issues.  I believe some of the reason is because students were confused with what the parentheses meant, while others needed a visual model to make a better connection.

The class reviewed a few different examples and we went back to a concrete representation.  I find this is the place where solid understanding is developed before we  move to more abstract models.

Students have been use to using base-ten blocks, counters and unifix cubes to put together and take apart numbers.  Students were asked to use cubes to show an understanding of the distributive property.  They used a dice to create a multiplication problem and then split it into two parts.  They then wrote on the desks (who doesn’t love that?) to show multiple number models to indicate the total.

One issue was trying to figure out how to divide or create the partition.  Some students used dice to indicate where to split the model.

From here, students transitioned to problems involving larger numbers and in an abstract form.  They were more successful this time around.  Students then worked in groups to complete this OpenMiddle problem.  They worked in this task for about 15 minutes using whiteboards in the process.  This was a quality activity that has students trying out multiple numbers to make the equation work.

I’m looking forward to revisiting the distributive property when schools starts back up in January 2020.

Friday was our last day of school for about two weeks.  We’re officially on winter break, but it doesn’t feel like it yet.  I’m sure it’ll sink in on Monday when my alarm clock doesn’t go off at the normal time.

## Exploring the Distributive Property

My fifth grade crew has recently been exploring the distributive property.  What’s interesting is that most of the students have used the property before, but it just wasn’t labeled.  Students mentioned this as I introduced the concept earlier last week.  Although most of the students in the class had background knowledge of how to distribute numbers, the level of that understanding differs depending on the student. The majority of students have mastered many concepts related to number sense, but pre-algebra concepts are fairly new to them. That is one of the reasons that I chose to beef up a lesson related to the distributive property. I have a few specific resources related to teaching the distributive property that I thought might be helpful for this lesson.

The substantive mathematical idea of this month-long instructional unit is to have students experience algebra and use it with geometry/measurement ideas with algebra notation. Later on in the unit students explore the distributive property, apply order of operations, simplify expressions, solve equations, utilize the Pythagorean theorem and use size-change factors.

The lesson began with an agenda.  The mastery objective for the day was “students will be able to identify and use the distributive property to simply expressions.” I briefly explained and drew an example of the distributive property on the whiteboard. At this point I wanted students to get a quick overview of the distributive property in action. This quick overview seemed to help introduce the concept to students that haven’t seen it before.

Students were then placed in groups to complete an initial distributive property activity. A scenario was given where students were asked to purchase three gifts for three different grandkids. Each grandchild would receive the same items. Students were asked to supply number sentences.  Feel free to download the sheet here.

The groups presented their findings and number sentences. During this time I was able to showcase how the distributive property can be utilized in this scenario. Based on the responses, students were still having trouble identifying and using the distributive property. Also, I was finding that students were adding each individual number instead of using a more efficient distributive property. Seeing that students needed more practice opportunities, I decided to move on to a rectangle method activity.

Students were then asked to find the area of the above rectangle using two different numbers sentences. I chose this particular assignment because of the math connection opportunities. Students were recently studying measurement concepts during the last unit and it’s still fresh in their minds. So, students were given rectangles that were split into segments and they were asked to show different number sentences find the area of the shaded portion.  I placed the page on the document camera and the class reviewed it together. Students were given time to reflect, make connections and ask questions during this time. I also gave students an opportunity to preview the next few lessons and see how understanding the distributive property will help them as they simplify expressions later in the week.

The distributive property activity contributes to the students’ developmental conceptual understanding of the mathematical idea. Students are asked to create a rectangle, divide it, and then use two different number sentences to showcase the shaded area. Students are using factoring strategies to group numbers in order to find the area. In doing this, students are acknowledging that the distributive property is evident in the combination process.

I believe there were challenges evident when I presented these mathematical ideas to the class. Students often come into class with preconceived notions that parentheses are only used during problems involving the order of operations. I believe that the students’ understanding of the distributive property was strengthened through the use of the rectangle area activity. Although their understanding seemed to improve, some students need to be guided through the activity. They were unsure of how to start the problem and some needed prompts.

I believe that the student work I collected suggests that the next step in my instruction is to expand on being able to use the distributive property and combine it with translating equations into expressions.  The next sequential step is to use equations to solve problems involving integers. Although students have used integers in the past, it may be beneficial to review how negative integers impact the distributive process. Also, as I gave students feedback, I wondered if they would’ve been able to complete the same number sentences, but distribute the numbers from both sides of the parentheses. For example, could they connect that 5(11 – 10) is the same as (11-10)5 ? They’ve only encountered the first example, so this may be something worth investigating for the next time I plan this lesson sequence.  Having practice with these types of problems will benefit students, as they need to have experiences with using signed numbers with expressions.